1. Introduction
Consider the nonhomogeneous and nonlinear Dirichlet boundary value problem:
where
is a bounded open domain of
(
) and
is a Leray-Lions operator defined from the weighted Sobolev spaces with variable exponent
into its dual
with
and
. The datum
is a measure that admits an L1-dual composition.
Throughout the paper, we suppose that the exponent
is an element of
= {log-Hölder continuous function
such that
} (where for all
, we denote
and
by
and
) and that
is a weight function defined on
(i.e.,
is a measurable function which is strictly positive a.e. in
) satisfying:
(1.1)
(1.2)
(1.3)
The problem
is studied where the following assumptions are satisfied:
(H1) a is a Carathéodory function satisfying:
(1.4)
(1.5)
(1.6)
where
is a positive function in
,
is a continuous function and
are strictly positive constants.
(H2) The second member
is supposed of the form:
(1.7)
where
and
.
A typical example of the problem
is the following involving the so-called
-Laplacian operator with weight:
The operator
becomes p-Laplacian when
(a constant) and
. The
-Laplacian operator with weight possesses more complicated nonlinearities than the classical p-Laplacian, for example, it is inhomogeneous with some degeneracy or singularity. For the applied background of
-Laplacian, we refer to (see [1] ). The study of differential equations with variable exponents has been a very active field in recent years, we find applications in electro-rheological fluids (see [1] and [2] ) and in image processing (see [3] ).
Under our assumptions (in particular (1.5), the problem
does not admit, in general, a weak solution since the term
may not belong to
. To overcome this difficulty we use in this paper the framework of L1-version of Minty’s lemma (similar to the one used in [4] ). And due to the assumption (1.6) it may be a degenerated or singular problem. Note also that, since the datum is a measure, then the notion of a weak solution cannot be used, hence it is replaced by another approach of solution calling
-solution (see definition 3.1 below).
Dirichlet problem of type
was considered in ( [5] [6] ), where in the first work the case of
(a constant) and
is treated, while the second work concerns the degenerated case with
(a constant). Hence our present paper can be seen as a generalization of the two works ( [5] [6] ). We also point out that the existence of solutions for elliptic equations with variable exponents can be found in [7] [8] and [9] and.
This paper is divided into three sections, organized as follows: In Section 2, we introduce and prove some properties of the weighted Sobolev spaces with variable exponent and in Section 3, we prove the existence of
-solutions of our problem
. Among the research objectives of this article is to introduce it for applications in physics and also will be a platform for the problem systems of Dirichlet and others.
2. Weighted Sobolev Spaces with Variable Exponent
Let
and
be a weighted function in
.
We define the weighted Lebesgue space with variable exponents
as the set of all measurable functions
for which the convex weight-modular
is finite. The expression
defines a norm in
, called the Luxemburg norm.
Proposition 2.1. The space
is a Banach space.
Proof. By considering the operator
defined by
for all
, it’s easy to show that
is an isomorphism and hence we can deduce.
Remark 2.1. When
, the weighted Lebesgue spaces with variable exponent
coincides with the Lebesgue space with variable exponent
.
The weight-modular
coincides with the modular
defined on
by
(for more details see [10] [11] [12] and [13] ).
Lemma 2.1. For all function
, the following assertions are satisfied:
1)
, respectively.
2) If
, then
.
3) If
, then
.
Proof. It suffices to remark that
and
, and using the analogous result in [13].
Proposition 2.2. Let
be a bounded open domain of
and
be a weight function on
satifying the integrability condtions (1.1) and (1.2). Then
↪
.
Proof.
Let K be an included compact on
. By vertue of Hölder inequality we have,
Hence, the conditions (1.1) and (1.2) allow to conclude.
We define the weighted Sobolev space with variable exponents denoted
, by
equipped with the norm
which is equivalent to the Luxemburg norm
Proposition 2.3. Let
be a weight function on
satisfying the conditions (1.1) and (1.2). Then the space
is a Banach space.
Proof. Let
be a Cauchy sequence in
. Then
is a Cauchy sequence in
and
is also a Cauchy sequence in
for all
. By vertue of proposition 2.1, we can deduce that there exist
and
such that:
and
Moreover, by using proposition 2.2, we have
. Thus, for all
one has,
Hence
, i.e.
.
Consequently,
and
Remark 2.2. Since
satisfies the conditions (1.1) and (1.2), it’s easy to prove that
is included in
; then we can define the following space
which is also a Banach space under the norm
.
Proposition 2.4. (Characterization of the dual space).
Let
and
be a weight function on
satisfying the conditions (1.1) and (1.2). Then for all
, there exists a unique system of functions
such that,
Proof. The proof of this proposition is similar to that used in [12] (theorem3.16).
Now, let us introduce the function
defined by
We have
and
Proposition 2.5. Let
and
be a weight function on
which satisfies the conditions (1.1), (1.2) and (1.3). Then
↪
.
Proof. According to the Hölder inequality and the condition (1.3), one has
If we take
, we then obtain
where
Consequently, we can write
where
and
Thus
(2.1)
Note that
denotes some positive constant which may be changing step by step.
Since
p.p. in
, then, there exists a positive constant C such that
Thus, we conclude that
↪
.
Corollary 2.1. Let
and
be a weight on
which satisfies the conditions (1.1), (1.2) and (1.3). Then
↪↪
, for
.
Corollary 2.2. Let
and
be a weight function on
which satisfies the conditions (1.1), (1.2) and (1.3). Then
Proof. Let
. Since
, we deduce by vertue of the embedding
↪
that,
Thus, in view of the proposition 2.5, we obtain
which allows to conclude that
3. Existence Result
Consider the nonhomogeneous nonlinear Dirichlet boundary problem:
Definition 3.1. A function u is called a
-solution of problem
if:
Theorem 3.1. Let suppose that the assumptions (1.1)-(1.7) are satisfied. Then the problem
has at least one
-solution.
Remark 3.1. Note that in the particular case where
(constant),
and
, the same result is proved in [14] by using the approach of pseudo-monotonicity.
3.1. Approximate Problem
Let
be a sequence of functions in
which converges strongly to f in
such that
. For
, we consider the approximate problem of
This section is devoted to establishing the existing solution for the approximate problem
.
Theorem 3.2. The operator
defined by,
is bounded, coercive, hemicontinuous and pseudo-monotone.
Proof of Theorem 3.2
● The operator
is bounded. Indeed for all
, one has
Since
is continuous and
a.e. in
, then
is bounded in
; hence the operator
is bounded.
● The operator
is hemicontinuous. Indeed, let t be a reality that tends to
. We have
Since
is bounded in
, we deduce that
converges to
weakly in
as t tends to
.
● The operator
is coercive. Indeed, for all
, we have
where
Obviously, we have
tends to infinity, when
, hence we conclude.
● It remains to show that
is pseudo-monotone: Let
be a sequence in
such that
(3.1)
Firstly, we prove that
converges to
weakly in
. Indeed, since
is a bounded sequence in
, then by the growth condition,
is bounded in
, therefore there exists a function
such that,
(3.2)
Hence, we can write
(3.3)
On the one hand, by (1.5), we have
Then
(3.4)
Since
strongly in
and a.e. in
, then
(3.5)
Therefore,
(3.6)
and
(3.7)
By vertue of (3.2), we have
(3.8)
Now, combining (3.4)-(3.6) and (3.7), we obtain
Due to (3.3), we deduce that
This implies that,
(3.9)
On the other hand, choose
in (3.9) (with
). It’s easy to see that
Hence
, and we deduce that
weakly converges to
in
.
Secondly, we prove that
. Indeed, in view of (3.2) and (3.3), we have
It remains to show that,
For that, we have
Since
, we deduce that
Therefore,
Hence,
. This achieved the proof.
3.2. Proof of Theorem 3.1
The proof is divided into 4 steps.
Step 1: We will show that
is a Cauchy sequence in measure. Using
as a test function in
leads to,
From (1.6) and (1.7), we deduce for all
that,
Now, by Young’s inequality, we obtain
(3.10)
Then, one has
for
, which implies that
(3.11)
Let
large enough and
be a ball of
. Using (3.11) and applying Hölder’s inequality and Poincaré’s inequality, we obtain
(by vertue of Corollary 2.2) (3.12)
(by vertue of Lemma 2.1)
where
which implies that,
(3.13)
So, we have, for all
,
(3.14)
Since
is bounded in
, there exists a subsequence, still denoted by
and a measurable function
such that
converges to
weakly in
, strongly in
and
almost everywhere in
. Hence
is a Cauchy sequence in measure in
.
Let
. Then by (3.13), there exists
such that,
This proves that
is a Cauchy sequence in measure in
, thus converges almost everywhere to some measurable function u. Hence
(3.15)
Step 2: We shall prove that
(3.16)
Let
and let n be large enough (
). Using the admissible test function
in
leads to
(3.17)
i.e.,
(3.18)
which implies that
(3.19)
Thanks to assumption (1.5) and the definition of truncation function, we have
(3.20)
Combining (3.19) and (3.20), we obtain (3.16).
Step 3: We claim that
(3.21)
Let
. Since
converges to
weakly in
, then
(3.22)
Thanks to assumption (1.4), we have
(3.23)
where
and C is a positive constant. Since
converges to
weakly in
, strongly in
and a.e. in
, thus
and
Combining (3.21), (3.22) and using Vitali’s theorem, we obtain
(3.24)
Now, we show that
(3.25)
In the first time, we have
a.e in
,
and
in
. In the second time, by using Vitali’s theorem we obtain (3.25).
Since
, one has
(3.26)
Thanks to (3.24), (3.25) and (3.26), we obtain (3.21).
Step 4: In this step, we introduce the following generalization of Minty’s lemma in weighted Sobolev space with variable exponents
(which is proved in [15] ).
Lemma 3.1. ( [15] ) Let u be a measurable function such that
for every
. Then the following statements are equivalent:
1)
2)
for every
and for every
.
Finally, the result (3.21) and the lemma 3.1 lead to the completion of the proof of theorem 3.1.
4. Conclusion
In this article, we have demonstrated the existence of a solution of a problem with a second measure member and in the space of Sobolev with variable exponent using Minty’s lemma. It is a very important technique in which we use the notions of hemicontinuous and pseudo-monotonic instead of broad or strict monotony.